Landauer's erasure principle in non-equilibrium systems

In two recent papers, Maroney and Turgut separately and independently show generalisations of Landauer's erasure principle to indeterministic logical operations, as well as to logical states with variable energies and entropies. Here we show that, although Turgut's generalisation seems more powerful, in that it implies but is not implied by Maroney's and that it does not rely upon initial probability distributions over logical states, it does not hold for non-equilibrium states, while Maroney's generalisation holds even in non-equilibrium. While a generalisation of Turgut's inequality to non-equilibrium seems possible, it lacks the properties that makes the equilibrium inequality appealing. The non-equilibrium generalisation also no longer implies Maroney's inequality, which may still be derived independently. Furthermore, we show that Turgut's inequality can only give a necessary, but not sufficient, criteria for thermodynamic reversibility. Maroney's inequality gives the necessary and sufficient conditions.